Anscombe & Aumann Expected UtilityBetting and Insurance

Econ 2100 Fall 2017

Lecture 11, October 3

Outline

1 Subjective Expected Utility2 Qualitative Probabilities3 Allais and Ellsebrg Paradoxes4 Utility Of Wealth5 Insurance and Betting

Anscombe and Aumann Structure

Ω = 1, 2, . . . ,S is a finite set of states, with generic element s ∈ Ω.

X is a finite set of size n of consequences with a generic element x ∈ X .∆X is the set of all probability distributions over X .

Elements of Ω reprent subjectively perceived randomness, while elements of∆X represent objectively perceived randomness.

Preferences are on H = (∆X )Ω, the space of all functions from Ω to ∆X .

An act is a function h : Ω→ ∆X that assigns a lottery in ∆X to each s ∈ Ω .

Let hs = h(s) ∈ ∆X , and denote hs (x) = [h(s)](x) ∈ [0, 1].This is the probability of x conditional on s , given the act h: (Pr(x |s , h))

The set of constant acts is

Hc = f ∈ H : f (s) = f (s ′) for all s ∈ ΩIf f , g ∈ H, then the function αf + βg : Ω→ ∆X is defined by

[αf + βg ](s) = αf (s) + βg(s).

Draw a picture to make sure you see how this works.

Subjective Expected Utility (SEU): IdeaStarting from preferences, identify a probability distribution µ ∈ ∆Ω and autility index v : X → R such that a utility representation of these preferences is

U(h) =∑s∈Ω

µ(s)

[∑x∈X

v(x)hs (x)

]equivalently:

f % g ⇔∑s∈Ω

µ(s)

[∑x∈X

v(x)fs (x)

]≥∑s∈Ω

µ(s)

[∑x∈X

v(x)gs (x)

]

Some Accounting Details

U(h) is the (subjective) expected value of v given act h since

∑s

µ(s)

[∑x

v(x)hs (x)

]︸ ︷︷ ︸

Eh(s)[v (x)]

=∑x

v(x)

∑s

Pr(s)︷︸︸︷µ(s)

Pr(x |s,h)︷ ︸︸ ︷hs (x)

︸ ︷︷ ︸

Pr(x)

.

∑s µ(s)hs (x) is the total or unconditional subjective probability of x under the

function h, denoted Pr(x).

Therefore Pr(x) =∑s Pr(s)Pr(x |s , h), since µ(s)hs (x) = Pr(s)Pr(x |s , h).

Last Class: State Dependent Expected Utility

TheoremThe preference relation % on H is complete, transitive, independent andArchimedean if and only if there exists a set of vNM indices v1, . . . , vS : X → Rsuch that. the utility representation is

U(h) =∑s∈Ω

∑x∈X

vs (x)hs (x)

This theorem does not define a unique probability distribution over Ω (seetoday’s homework).

To pin down a probability distribution one needs one more assumption:

The binary relation % on H is state-independent if, for all non-null statess, t ∈ Ω, for all acts h, g ∈ H, and for all lotteries π, ρ ∈ ∆X ,

(h−s , π) % (h−s , ρ) ⇒ (g−t , π) % (g−t , ρ).

Subjective Expected Utility TheoremTheorem (Expected Utility Theorem, Anscombe and Aumann)

A preference relation % on H is complete, transitive, independent, Archimedean,and state-independent if and only if there exists a vNM index v : X → R and aprobability µ ∈ ∆Ω such that

U(h) =∑s∈Ω

µ(s)

[∑x∈X

v(x)hs (x)

]is a utility representation of %.Moreover, this representation is unique up to affi ne transformations provided atleast two acts can be ranked strictly.

Therefore:

h % g ⇔∑s∈Ω

µ(s)

[∑x∈X

v(x)hs (x)

]≥∑s∈Ω

µ(s)

[∑x∈X

v(x)gs (x)

]The second part says: if there exist h, g ∈ H such that h g and U(h)defined above represents %, then U ′(h) =

∑s µ′(s) [

∑x v′(x)hs (x)] also

represents % if and only if µ′ = µ and v ′ = av + b for some a > 0 and b ∈ R.

RemarkThe probability distribution µ is unique.

Subjective Expected Utility TheoremTheorem (Expected Utility Theorem, Anscombe and Aumann)

A preference relation % on H is independent, Archimedean, and state-independentif and only if there exists a vNM index v : X → R and a probability µ ∈ ∆Ω suchthat

U(h) =∑s∈Ω

µ(s)

[∑x∈X

v(x)hs (x)

]=∑x∈X

v(x)

∑s∈Ω

Pr(s)︷︸︸︷µ(s)

Pr(x |s)︷ ︸︸ ︷hs (x)

︸ ︷︷ ︸

Pr(x)

is a utility representation of %. This representation is unique up to affi netransformations.

Preferences identify two things:the utility index over consequences v : X → R andthe probability distribution over states µ ∈ ∆Ω.

Different preferences may imply different beliefs µ on Ω.von Neumann—Morgenstern’s Theorem only identifies the utility index v .

For this reason, this is called Subjective Expected Utility.

Anscombe & Aumann Proof

We will not do it in detail, but here is a breakdown of what would happen.

Proof.

First, convert each act h to a vector x ∈ [0, 1]S , by assigning each dimensions ∈ 1, ...,S a vNM expected utility for hs

this gives[∑

x v (x)hs (x)]in each state s ;

monotonicity, a consequence of state-independence, is essential for this step.

Then, construct an independent and Archimedean preference relation on[0, 1]S and the measure µ is the dual of the affi ne utility on [0, 1]S .

The necessity and uniqueness parts are as usual.

Notice that this argument involves using the mixture space theorem (or vNMtheorem) twice.

the first time to find[∑

x v (x)hs (x)]in each state s (this is Eh(s)[v (x)]);

the second to find the measure µ as the dual of the affi ne utility on [0, 1]S .

Savage Subjective Expected UtilityAnscombe & Aumann assume the existence of an objective randomizing device(preferences are defined over functions from states to objective lotteries).

∆X is a convex space and one can apply the mixture space theorem.

Savage’s theory instead considers all uncertainty as subjective: no probabilitiesare assumed a priori; all probabilities are identified by preferences over basicobjects.

This comes at the cost of higher mathematical complexity (see Kreps’Theoryof Choice).

Ω is the state space, and X is a finite set of (sure) consequences.

Savage acts are elements of XΩ, the space of all functions from Ω to X .

The decision maker has preferences % over these acts.The expected utility representation is something like

U(f ) =∑x∈X

v(x)µ(f −1(x)) =

∫Ω

v fdµ = Eµ[v f ],

where f is any act, v : X → R is a vNM utility index on consequences, and µis a probability measure on Ω.

Utility and probability are fully endogenous.

Probability Distributions

Economists think of preference as primitive, and derive subjective probabilityas a piece of a subject’s utility function.

Some statisticians think about probabilities the same way economists thinkabout utility functions.

Definition

A (finitely additive) probability measure on Ω is a function µ : 2Ω → [0, 1] suchthat:

1 µ(X ) = 1;2 If A ∩ B = ∅, then µ(A ∪ B) = µ(A) + µ(B).

They usually assume that µ is also countably additive, which is very usefultechnically.

Qualitative Probabilities

Definition

A binary relation ∗ on 2Ω is a qualitative probability if:

1 ∗ is complete and transitive;2 A ∗ ∅, for all A ⊆ Ω;3 Ω ∗ ∅;a4 A ∗ B if and only if A ∪ C ∗ B ∪ C , for all A,B,C ⊆ Ω such thatA ∩ C = B ∩ C = ∅.

aBy A ∗ B , we mean A ∗ B and not B ∗ A, i.e. ∗ is the asymmetric component of ∗.

Think of this as a binary relation ∗ expressing the idea of “more likely than”.This is a “personal” characteristic; it varies from individual to individual andcan be use to infer the existence of “personal probabilities”.

Qualitative Probabilities and Probability MeasuresThe relationship between qualitative probability relations and probabilitymeasures is similar to the one between preference orders and utility functions.

Definition

The probability measure µ represents the binary relation ∗ on 2Ω if:

µ(A) ≥ µ(B) if and only if A ∗ B.

ResultAll binary relations that are represented by a probability measure arequalitative probabilities.

The converse is false: not all qualitative probabilities can be represented by aprobability measure.

A stronger version of additivity is required to guarantee representation by aprobability measure (see Fishburn 1986 in Statistical Science).

Our theorems about preferences infer the existence of probabilities. Theirtheorem about probabilities infer the existence of a preference relation.

Allais ParadoxThe decision maker must choose between the following objective lotteries.

and between

One reasonable prefrence may rank A B and D C .Any such preferences violates independence:

Consider the following lotteries E and F :

0.11A+ .89A = A B = 0.11E + .89A

0.11A+ .89F = C ≺ D = 0.11E + 0.89F

But independence says for all f , g , h ∈ H and α ∈ (0, 1),f % g ⇔ αf + (1− α)h % αg + (1− α)h.

Ellseberg ParadoxAn urn contains 90 balls. 30 balls are red (r) and 60 are either blue (b) or yellow(y). One ball will be drawn at random. Consider the following bets.

and

Many prefer A to B and C over D. Is this consistent with expected utility?By expected utility, there exists a (subjective) probability distribution µ overr , y , b.From A B and C D we conclude:

µ(r)U($100) + (1 − µ(r))︸ ︷︷ ︸µ(b)+µ(y )

U($0) > µ(b)U($100) + (1 − µ(b))︸ ︷︷ ︸µ(r )+µ(y )

U($0)

⇒ µ(r) > µ(b)and(µ(b) + µ(y ))U($100) + µ(r)U($0) > (µ(r) + µ(y ))U($100) + µ(b)U($0)

⇒ µ(r) < µ(b)

This contradicts the hypothesis of consistent subjective beliefs.

(Knightian) Uncertainty vs. RiskRisk

B R

B = 30 R = 70

B+R=100

P(B) = 310 P(R) = 7

10

Knightian Uncertainty (Ambiguity)

B R

20 ≤ B ≤ 50 50 ≤ R ≤ 80

B+R=100

P(B) =? P(R) =?

Some theories that can account for Knightian Uncertainty.

Gilboa & Schmeidler (weaken independence): Maxmin Expected Utility

x % y ⇔ minµ∈C

∑s

µ(s)u(xs ) ≥ minµ∈C

∑s

µ(s)u(ys )

Truman Bewley (drop completeness): Expected Utility with Sets

x % y ⇔∑s

µ(s)u(xs ) ≥∑s

µ(s)u(ys ) for all µ ∈ C

Probability Distribution On WealthMany applications of expected utility consider preferences over probabilitydistribution on wealth (a continuous variable).A probability distribution is characterziedd by its cumulative distributionfunction.

Definition

A cumulative distribution function (cdf) F : R→ [0, 1] satisfies:

x ≥ y implies F (x) ≥ F (y) (nondecreasing);

limy↓x F (y) = F (x) (right continuous);a

limx→−∞ F (x) = 0 and limx→∞ F (x) = 1.

aRecall that, if it exists, limy↓x f (y ) = limn→∞ f (x + 1n ).

Notation

µF denotes the mean (expected value) of F , i.e. µF =∫x dF (x).

δx is the degenerate distribution function at x ; i.e. δx yields x with certainty:

δx (z) =

0 if z < x

1 if z ≥ x.

Expected Utility Of WealthAs usual, the space of all distribution functions is convex and one can definepreferences over it.The utility index v : R→ R is defined over wealth (can be negative).The expected utility is the integral of v with respect to F∫

v(x)dF (x) =

∫vdF

If F is differentiable, the expectation is computed using the density f = F ′:∫v dF =

∫v (x)f (x) dx .

von Neumann and Morgenstern Expected UtilityUnder some axioms, there exists a utility function U on distributions defined asU(F ) =

∫v dF , for some continuous index v : R→ R over wealth, such that

F % G ⇐⇒∫v dF ≥

∫v dG

Axioms not important from now on (need a stronger continuity assumption).We always think of v as a weakly increasing function (more wealth cannot bebad).

Simple Probability

A simple probability distribution π on X ⊂ R is specified bya finite subset of X called the support and denoted supp(π), andfor each x ∈ X , π(x) > 0 with

∑x∈supp(π) π(x) = 1

If we restrict attention to simple probability distributions, then even if X isinfinite, only elements with strictly positive probability count.

The utility index v : X → R is defined over wealth (can be negative).The expected utility is the expected value of v with respect to π∑

x∈supp(π)

π(x)v(x)

One can write more money is better as: for each x ,y ∈ X such that x > ythen δx δy .We can use this setting to think about many applied choice under uncertaintyproblems like betting and insurance.

BettingA Gamble

Suppose an individual is offered the following bet:win ax with probability p lose x with probability 1− p

The expected value of this bet is

pax + (1− p) (−x) = [pa+ (1− p) (−1)] x

Definition

A bet is actuarially fair if it has expected value equal to zero (i.e. a = 1−pp ); it is

better than fair if the expected value is positive and worse than fair if it is negative.

How does she evaluate this bet? Use the expected utility model to find out

If vNM index is v(·) and initial wealth is w , expected utility is:probability of winning

p v (w + ax)utility of wealh if win

+probability of losing

(1− p) v (w − x)utility of wealth if lose

How much does she want of this bet? Answer by finding the optimal x .

Betting and Expected Utilitywin ax with probability p lose x with probability 1− p

The consumer solves

maxxpv (w + ax) + (1− p) v (w − x)

The FOC ispav ′ (w + ax) = (1− p) v ′ (w − x)

rearrangingpa

(1 − p)=v ′ (w − x)

v ′ (w + ax)

If the bet is fair, the left hand side is 1. Therefore, at an optimum, the righthand side must also be 1.

If the vNM utility function is strictly increasing and strictly concave (v ′ > 0and v ′′ < 0), the only way a fair bet can satisfy this FOC is to solve

w + ax = w − xwhich implies x = 0.

She will take no part of a fair bet.

What happens with a better than fair bet?

InsuranceAn Insurance Problem

An individual faces a potential “accident”:

the loss is L with probability π nothing happens with probability 1− π

DefinitionAn insurance contract establishes an initial premium P and then reimburses anamount Z if and only if the loss occurs.

DefinitionInsurance is actuarially fair when its expected cost is zero; it is less than fair whenits expected cost is positive.

The expected cost (to the individual) of an insurance contract is

P −[π(−Z )

loss+ (1− π) (0)

no loss

]= P − πZ

Fair insurance meansP = πZ

Insurance and Expected UtilityAn Insurance Problem

An individual with current wealth W and utility function v(·) faces a potentialaccident:

lose L with probability π or lose zero with probability 1− π

If she buys insurance, her expected utility is

πv(W − L− P + Z︸ ︷︷ ︸)wealth if loss

+ (1− π) v( W − P︸ ︷︷ ︸wealth if no loss

)

For example, if the loss is fully reimbursed (Z = L), this becomesπv (W − P) + (1 − π) v (W − P) = v (W − P)

Will she buy any insurance? Yes ifπv (W − L− P + Z ) + (1− π) v (W − P)︸ ︷︷ ︸

expected utility with insurance

≥ πv (W − L) + (1− π) v (W )︸ ︷︷ ︸expected utility without insurance

How much coverage will she want if she buys any coverage?Find the optimal Z .

The answer depends on the premium set by the insurance company P (whichcould depend on Z ) as well as the curvature of the utility function v .

Problem Set 6 (Incomplete)Due Tuesday 9 October

1 Suppose Ω = s1, s2, s3 and X = x1, x2, x3. The preference relation % is independent, Archimedean, andstate-independent. The following are known: δx1 δx2 δx3 and(δx1 , δx2 , δx3) ∼ (δx1 , (0,

12 ,

12 ), (0, 12 ,

12 )) ∼ (( 23 ,

13 , 0), ( 23 ,

13 , 0), δx3). Find the decision maker’s subjective belief µ.

2 Suppose Ω = s1, s2, s3 and X = x1, x2. Assume % is a preference relation on (∆X )Ω which admits an

Anscombe-Aumann expected utility representation. Denote acts as h =

(hs1(x1), hs1(x2))(hs2(x1), hs2(x2))(hs3(x1), hs3(x2))

. The following areknown: (1, 0)

(1, 0)(1, 0)

(0, 1)

(0, 1)(0, 1)

;

(1, 0)(0, 1)(0, 1)

%(0, 1)

(1, 0)(0, 1)

;

(1, 0)(0, 1)(0, 1)

∼(0, 1)

(0, 1)(1, 0)

.What is the set of beliefs in ∆Ω that is consistent with these data (if the decision maker followsAnscombe—Aumann expected utility)? Is a unique belief identified, i.e. is the identified set a singleton? What ifthe middle weak preference were an indifference relation?

3 An individual has expected utility preferences with utility over money given by some twice differentiable function vwith v ′ > 0 and v ′′ < 0. She can choose the value of t in the following “bet”: receive txs dollars with probabilityπs with s = 1, ...,S (note that some of the xs are negative). Her initial wealth is w for all s.

1 Show that∑S

s=1 πsxs = 0 if and only if the optimal t is equal to zero.2 What (if anything) can you say about the optimal t when

∑Ss=1 πsxs > 0? How about

∑Ss=1 πsxs < 0?

3 Interpret these results in terms of better than fair gambles. What would happen if v ′′ = 0?

4 An individual whose wealth is W > 0 could have an accident in which she loses L with probability π. Let u denotethe individual utility for money, and assume this function is differentiable at least twice with u′ > 0 and u′′ < 0.An insurance policy specifies the premium the consumer has to pay regardless of the accident (P ), and theamount the insurance company reimburses if the accident occurs (Z ). The latter is chosen by the consumer.

1 Show that if insurance is actuarilly fair, the consumer will insure the entire loss.2 Characterize the relation between the optimal insurance coverage and the premium P . Interpret this result. (HINT:draw a picture)